For a rarefied gas, \(\beta \ll 1\) , the phases \(x_i^0\)  can be assumed to follow Poisson distribution on \(\mathbb {R}\)  with density \(\beta \) . In such a gas, the total spatial shift of a “tracer” soliton with spectral parameter \(\eta =\eta _1\)  (we shall call it \(\eta _1\) -soliton) over the time interval \(\mathrm {d}t\) , due to the interactions with \(\mu \) -solitons, \(\mu \in \Gamma \) , is given by \(\Delta _1 \approx \int _\Gamma [\Delta (\eta _1, \mu ) |s_0(\eta _1) - s_0(\mu )| f(\mu ) \mathrm {d}\mu ] \mathrm {d}t\) , where \(s_0(\eta )=4\eta ^2\)  is the speed of a free, non-interacting soliton. This simple argument was used by Zakharov in 1971 [1]}.
